PIBNet: a Physics-Inspired Boundary Network for Multiple Scattering Simulations

TL;DR

PIBNet tackles the computational bottleneck of solving boundary integral equations in exterior multiple scattering by learning the boundary trace via a physics-informed, multiscale graph neural network. The method explicitly models long-range obstacle interactions through a sparse, physics-guided distant-edge graph and processes information across multiple graph levels to predict the boundary solution, enabling rapid reconstruction of the volumetric field via the boundary integral representation. On a new benchmark of Laplace and Helmholtz problems with multiple disjoint obstacles, PIBNet outperforms state-of-the-art learning-based PDE solvers and demonstrates stronger generalization to configurations with more obstacles. The approach offers a practical, scalable alternative to traditional BEM solvers for parametric studies and real-time simulations in wave propagation problems.

Abstract

The boundary element method (BEM) provides an efficient numerical framework for solving multiple scattering problems in unbounded homogeneous domains, since it reduces the discretization to the domain boundaries, thereby condensing the computational complexity. The procedure first consists in determining the solution trace on the boundaries of the domain by solving a boundary integral equation, after which the volumetric solution can be recovered at low computational cost with a boundary integral representation. As the first step of the BEM represents the main computational bottleneck, we introduce PIBNet, a learning-based approach designed to approximate the solution trace. The method leverages a physics-inspired graph-based strategy to model obstacles and their long-range interactions efficiently. Then, we introduce a novel multiscale graph neural network architecture for simulating the multiple scattering. To train and evaluate our network, we present a benchmark consisting of several datasets of different types of multiple scattering problems. The results indicate that our approach not only surpasses existing state-of-the-art learning-based methods on the considered tasks but also exhibits superior generalization to settings with an increased number of obstacles. github.com/ENSTA-U2IS-AI/pibnet

Figures (12)

• Figure 2: Illustration of some directed graph representations used in PIBNet in the case of two obstacles. Graph edges are in dark blue, while the light blue corresponds to the obstacle meshes. $\mathcal{G}^0$: the boundary graph (arrows have been omitted for readability), $\mathcal{G}^{1\rightarrow 2}$: the downsampling graph from level 1 to 2, $\mathcal{G}^2$: the distant nodes graph and $\mathcal{G}^{2\rightarrow 1}$: the upsampling graph from level 2 to 1. We omit the downsampling and upsampling graphs $\mathcal{G}^{0\rightarrow 1}$ and $\mathcal{G}^{1\rightarrow 0}$ between levels 0 and 1 which are similar to graphs $\mathcal{G}^{1\rightarrow 2}$ and $\mathcal{G}^{2\rightarrow 1}$ at a lower resolution.
• Figure 3: Illustration of the processor part of PIBNet GNN architecture with a number of levels $L=3$. The graph representation used by each processor block is given above the corresponding block. It is composed of $N_b$ Boundary Graph Processor Blocks at the beginning, then $2$ Downsampling Blocks, $N_d$ Distant Nodes Graph Processor Blocks, $L-1$ Upsampling Blocks, and $N_b$ Boundary Graph Processor Blocks at the end.
• Figure 4: Qualitative results for the three problems of our benchmark with three obstacles (in white). From left to right are the 3D volumetric solution of the total field obtained with GMRES (the groundtruth), with PIBNet and with PTv3 predictions of the trace solution, respectively, and the corresponding errors relative to the groundtruth with PIBNet and PTv3, respectively. For each problem, the 3D volumetric solutions of the total field and their associated errors are sampled within a square domain of side length 10 on the plane $z=0$.
• Figure 5: Estimation errors as a function of the number of obstacles for the Laplace Dirichlet problem (left) and the Helmholtz Dirichlet problem (middle and right).
• Figure 6: Comparison of runtime on with respect to the number of obstacles for learning-based methods and for the BEM considering different different convergence tolerance thresholds $\mathrm{rtol}$ for GMRES.